hypothetical AA question

[brick]

Scenario:
-limit holdem, 4 bet cap, table is an average Mid Limit table with a few players who will call 2 bets but not 3
-UTG raises, you are next to act with AA
-You expect UTG to autobet the flop

Catch:
You can vary the amount of money you choose to play this hand with.

Could you increase your EV by sitting with 3 bb, calling preflop, and raising utg's auto bet, before going all-in?
(as opposed to 3-betting with a full stack of chips)

This would enable you to exploit the large preflop and flop edge that AA has by encouraging many callers preflop, while reducing the reverse implied odds because you will be all-in on the river.

Hypothetical? More like a fantasy world. I don't know that I've ever seen a limit table where players would cold-call a 2-bet but not a 3. Build the pot, build the pot, build the pot. Get as much money in there as you can preflop.

Also, with AA if there have been a lot of limpers or overcalls, I want everyone rethinking their decision one more time as it comes around from UTG's cap-bet. I want to try to run off some suited/connected cards if at all possible. Ideally the only people who stay are the ones playing wired pairs.